====== Simple 1-State ======

\( \require{newcommand} \)
\( \newcommand{\diff}[2]{\frac{d{#1}}{d{#2}}} \)

We begin by considering the simple 1-state model.

===== Simplicity =====

We define //simplicity// by the following criteria:
  - The assumptions of [[wp>reaction kinetics]] apply.
  - All reactions have a total [[wp>order of reaction]] less than three.  There can be zero (nil), one (solo), or two (pair) reactants involved in a reaction.  We assume higher order reactions are negligibly rare or can be decomposed into sets of lower order reactions.
  - All reactions involve change to only one reactant.  The product set differs from the reactant set by exactly one individual.  We assume reactions with more than one reactant changing are composed of stages where one reactant changes at a time.
  - Every reactant/product represents a single individual in the population.  Bonded states are not allowed.

===== States =====

States refers to the number of distinguishable states of the individuals in the population.  A state may be a species or a member's gender or physiological state.  In this 1-state model individuals are completely indistinguishable.  There exists only one species and no distinction between members of the species.

===== Reactions =====

In the case of a simple 1-state system this means there are only five possible reactions:
\[
\begin{array}{rcll}
  \emptyset & \xrightarrow{\alpha} & N         & \text{(spontaneous creation)} \\
  N   & \xrightarrow{\beta}        & 2 N       & \text{(induced creation)} \\
  2 N & \xrightarrow{\gamma}       & 3 N       & \text{(pair-induced creation)} \\
  N   & \xrightarrow{\delta}       & \emptyset & \text{(spontaneous removal)} \\
  2 N & \xrightarrow{\epsilon}     & N         & \text{(induced removal)}.
\end{array}
\]

==== Spontaneous creation ====

Spontaneous creation occurs when an individual presents independently of the tracked population.  It could happen through //untracked stages// (eg. seeds) or //immigration// from an outside pool.

==== Induced creation ====

The simplest interpretation of induced creation is //asexual reproduction//.  Another possibility is an //immigration// process where individuals prefer to cluster and are drawn in by pre-existing individuals.

==== Pair-induced creation ====

Like induced creation but requires the interaction of a pair.  //Sexual reproduction// would be an example, as would a more subtle //immigration// driver.

==== Spontaneous removal ====

Spontaneous removal represents a process where an individual is removed from the population, either via //death// or //emigration//.

==== Induced removal ====

Likewise, induced removal could represent //death// or //emigration//, but it is brought about through the interaction with another individual.  The removed individual may have been driven out by //competition//.

===== Dynamics =====

The population density, \(n\), will behave as
\[
  \diff{n}{t} = \alpha + (\beta - \delta) n + (\gamma - \epsilon) n^2 .
\]

To understand the dynamics we should determine the equilibria, where \(\diff{n}{t}=0\).  There are two important cases, where either the rates of the pairwise (order 2) reactions are balanced or unbalanced:
\[
\begin{array}{rcll}
  \gamma & =    & \epsilon & \text{(pairwise balance)} \\
  \gamma & \neq & \epsilon & \text{(pairwise imbalance)}. \\
\end{array}
\]

==== Special case: Pairwise balance ====

If //pair-induced creation// and //induced removal// occur at the same rates (\( \gamma = \epsilon \)) then the dynamics simplify to
\[
  \diff{n}{t} = \alpha + (\beta - \delta) n.
\]

In this case the behaviour of the system is straight-forward: If \( \beta \geq \delta \) then the population diverges (non-biological), otherwise the population converges to
\[
  n^* = \frac{\alpha}{\delta - \beta} > 0 \; \text{if} \; \delta > \beta.
\]

==== Equilibrium & Stability ====

The more general case is when the two pairwise reactions are not precisely balanced (\( \gamma \neq \epsilon \)).  Then the dynamics follow
\[
  \diff{n}{t} = \alpha + \beta' n + \gamma' n^2 .
\]
which has equilibria at
\[
  n_\pm = \frac{-\beta' \pm \sqrt{\beta'^2 - 4 \alpha \gamma'}}{2 \gamma'}
\]
where
\[
\begin{array}{rll}
  \beta'  & \equiv \beta - \delta           & \text{(excess induced creation)} \\
  \gamma' & \equiv \gamma - \epsilon \neq 0 & \text{(excess pair-induced creation)}.
\end{array}
\]

=== Biological significance ===

If \( \beta'^2 < 4 \alpha \gamma' \) then the population has no equilibria -- it diverges (to \(+\infty\) since the condition requires \(\gamma'>0\)).  This is clearly a non-biological result since sustaining populations do not diverge.  Another mechanism (eg. predation) is expected to dominate as the population grows such that the above description is no longer adequate.  If we restrict ourselves to the simple 1-state model then we can assume by appeal to //biological significance// that
\[
  \beta'^2 \geq 4 \alpha \gamma'.
\]

One could argue that, if \(\gamma'>0\) then the population will diverge for large enough starting densities so this may also be non-biological.  On the other hand if the population density never actually reaches this regime of population explosion then the above reactions may be an adequate description of the dominant processes occurring.  In that case the dynamics //are// biologically significant and can't be neglected. 

By plotting \( \diff{n}{t} \) vs. \(n\) we find the following conditions are required for biological significance.  In both cases \(n=0\) is either a stable equilibrium or grows to an equilibrium at \(n_- > 0\):
^  \(\gamma'< 0\)\\ \(n_+\leq 0, n_-\geq 0\)  ^  \(\gamma'> 0, \beta'<0\)\\ \(n_-\geq 0, n_+> 0\)  ^
|         {{svg>down-leq0-geq0.svg}}          |              {{svg>up-geq0-gr0.svg}}               |

From the diagrams we can also discover the stability of the equilibria.  We find the following combination of parameters yield biologically significant population dynamics((Recall: in all cases \(\alpha \geq 0\).)).  Conditions necessary for population sustenance are indicated.
^  \(\gamma'\)  ^  \(\alpha\)  ^                         \(\beta'\)                          ^  \(n_-\)  ^  \(n_+\)  ^                  Stability                  ^  Sustainable  ^
|     \(-\)     |    \(0\)     |                            \(-\)                            |   \(0\)   |   \(-\)   | \(n_+ < 0\) unstable, \(n_- = 0\) stable    |      No       |
|     \(-\)     |    \(0\)     |                            \(0\)                            |   \(0\)   |   \(0\)   | \(n_- = 0\) stable((For \(n\geq 0\).))      |      No       |
|     \(-\)     |    \(0\)     |                            \(+\)                            |   \(+\)   |   \(0\)   | \(n_- > 0\) stable                          |      Yes      |
|     \(-\)     |    \(+\)     |                          \(-/0/+\)                          |   \(+\)   |   \(-\)   | \(n_+ < 0\) unstable, \(n_- > 0\) stable    |      Yes      |
|     \(0\)     |    \(+\)     |                            \(-\)                            |           |           | \(n^* = -\alpha/\beta' > 0 \) stable        |      Yes      |
|     \(0\)     |    \(0\)     |                            \(-\)                            |           |           | \(n^* = 0 \) stable                         |      No       |
|     \(+\)     |    \(0\)     |                            \(-\)                            |   \(0\)   |   \(+\)   | \(n_- = 0\) stable, \(n_+ > 0\) unstable    |      No       |
|     \(+\)     |    \(+\)     |  \(-\)((Requires \(\beta' < - 2\sqrt{\alpha \gamma'} \).))  |   \(+\)   |   \(+\)   | \(n_- > 0\) stable, \(n_+ > n_-\) unstable  |      Yes      |



====== Summary ======

We find a complete description of all biologically-relevant population dynamics from a simple, 1-state reaction kinetics formulation.

===== Biological significance =====

For biologically meaningful behaviour of the dynamics we require
\[
  \gamma < \epsilon
\]
or
\[
  \gamma \geq \epsilon, \delta - \beta > 2 \sqrt{\alpha (\gamma - \epsilon)}.
\]

In all other cases the population density diverges or can become negative.

===== Sustainability =====

An important subset of biologically significant behaviour is when the population is sustained, that is it can maintain a non-zero density indefinitely.  For this simple model this coincides with a stable equilibrium at some positive density, which can occur when((Assumes biological significance conditions are met.))
\[
  \alpha > 0
\]
or
\[
  \alpha = 0, \beta > \delta, \gamma < \epsilon.
\]